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2 q iq Rn n n )exp( ) , r r * ? ? (1) In equation (1), the symbol F q n ( ) r denotes the structure fac- tors of the individual scattering centres, r q the diffraction vector and r Rn the position vectors of the scattering centres. In the theory of the partial coherence of crystallites, the scattering centres are replaced by crystallites ;

it is as- sumed that the X-rays scattered by different crystallites can interfere. The structure factor of nanocrystallites, i.e. F q n ( ) r in equation (1), is a very broad function because its width is reciprocal to the crystallite size. For identical (very similar) crystallites having the same structure factor, equation (1) can be rewritten into the following form: ? Krystalografická spoleènost

68 D. Rafaja, V. Klemm, C. Wüstefeld, M. Motylenko, M. Dopita Figure 1. Microstructure model used for description of the X-ray scattering in nanocrystalline materials. Nearly defect-free nanocrystallites are separated by regions with extremely high de- fect density.

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315 qx qz Figure 2. Projection of the reciprocal lattices from two mutually disoriented fcc nanocrystallites into the qx-qz plane. The white circles belong to the first nanocrystallite, the grey circles to the second one. The numbers within the circles are the diffraction indices. I N F q N m w iq R m m m N ? + - * é ? ê ù ? ú = ? Re exp( r r r

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1 2 D (2) In equation (2), N is the number of diffracting crystal- lites and wm ?

01 , the degree of the partial coherence of crystallites with the distance D r Rm . The degree of the partial coherence is proportional to the overlap of their reciprocal lattice points;

wm =

0 for non-coherent crystallites, wm =

1 for fully coherent crystallites. In the classical kinematical diffraction theory, the interference term that is related to the partial coherence of crystallites, i.e. the sum in equation (2), is neglected. For overlapping reciprocal lattice points in nanocrystalline materials, the broad structure factor is multiplied by a harmonic function coming from the com- plex exponential function in equation (2), which frequency depends on the mean distance of the partially coherent crystallites. This multiplication causes a narrowing of the diffraction lines as shown in Fig. 3. The physical inter- pretation of this phenomenon is that the X-ray scattering cannot distinguish the crystallites with partially overlap- ping reciprocal lattice points from each other. Thus, the partially coherent nanocrystallites appear larger than they are. According t........